So, if you're going to ask this question of others you have to also examine yourself. I have been asking myself (and the Lord through prayer and reflection) if my preaching reflects the manner that I believe Jesus would. I try, but I know I fall short - I'm still learning.
What does it take to preach, much less communicate like Jesus? By this I mean the application of the question to go beyond the Sunday morning preaching of God's word. I mean how does this work out in our daily life. To our friends, co-workers, family members, and neighbors.
It should be obvious that you must first off know Jesus. Now, You can know a lot of data about Jesus and not know him. You could be a Jesus stalker! Do you really know Him? He said many controversial things and made some very daring claims. Do you really know HIm? Moreover, how much do you want to know him?
There is a church that my family once attended where the pastor never gave personal illustrations, but always canned responses and sermons more often based on books and studies that other pastors or writers had done. There have been times when my mother even accused him of essentially plagiarizing. It's one thing to strive to be like Jesus, it's another to just mimic someone else's style or steal their content just because they may be persuasive or engaging. If instead, you actually dig deep and get behind what Jesus said as the original authors wrote it in the context of history and the rest of scripture, you will have a much more authentic, impacting, powerful and reverent view of Jesus than often pervades our churches today. Present the real Jesus as close to how Jesus would present himself and you have the secret formula for growth - perhaps not in numbers, but in true spiritual depth.
Don't forget either that He can be seen reflected throughout the writings of scripture, both Old and New Testaments. It all points to HIM! Study well and you will see it! He speaks beyond the red letters. Listen and learn.
Finally, I would hope it would go without saying that communicating like Jesus means nothing unless you've actually been changed by His work for you on the cross. If that change doesn't take root in the depths of who you are and how you live, then you are entirely missing the point. To know HIm is love HIm, to love Him is to draw close to Him, to draw close to him is to grow to be like Him, to be like him is to love those he loved and serve them as he did. Words must meet action. Live like Him and your words will more accurately reflect Him. That's what we're called to do. That's what I want to do. I don't do it perfectly, but I'm not going to quit pursuing Him. How about you? Any thoughts on how we can do better at this together?
Some Scripture to consider:
Isaiah 42:1-3, Matthew 12:18-20